I would try a "fuse tap" which you can find at most auto parts stores. I have two that I bought at Advance Auto. If the link below doesn't work, go to the Advance web site and search on "fuse tap". These devices are rated at 10 amps maximum, but that's plenty for most applications unless you're running a lot of power. It's plenty for my 2m rig and my 50 watt HF rig, but check your CB radio's power requirements to be sure.
To use the fuse tap the way you want to use it, where the CB can only get power when the ignition key is on, go to your fuse box and find a circuit that is only powered when the key is on. Pull that fuse and put it in the appropriate slot on the fuse tap. Now put another fuse in your fuse tap, and this 2nd fuse will be the one that protects your CB. Now it's time to connect the hot wire to your CB. On the side of the fuse tap you'll see a red wire -- connect this to the hot wire going to your CB. Now plug the fuse tap into the fuse box from where you had pulled the fuse a few minutes ago. With the fuse tap in place, the circuit you selected is once again protected, plus you now have power to your CB.
A couple of things to note - if you don't get power to your CB, the fuse tap is installed upside down. Unplug it from the fuse box, flip it 180 degrees, and plug it back in. Also, be aware of current ratings. You don't want to exceed the rated capacity of the fuse tap, which is typically 10 amps (check the package to be sure). And as you already know, never replace a fuse with one that is made for a higher current rating than the one being replaced.
The other guys are right -- the best way to get your power is to go straight to the battery, with fuses on both the hot and ground connections. But for relatively low current applications of 10 amps or less, it's been my experience that the fuse tap works fine. I just hope you don't get ignition noise since you're using AM or SSB.
Good luck and 73!